Four-junction superconducting circuit

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by  Yueyin Qiu

Four-junction superconducting circuit
Yueyin Qiu,1,2 Wei Xiong,1,2 Xiao-Ling He,3 Tie-Fu Li,4,2 and J. Q. You2
1Department of Physics, Fudan University, Shanghai 200433,China
2Beijing Computational Science Research Center, Beijing 100193, China
3School of Science, Zhejiang University of Science and Technology, Hangzhou, Zhejiang 310023, China
4Institute of Microelectronics, Department of Microelectronics and Nanoelectronics and Tsinghua National
Laboratory of Information Science and Technology, Tsinghua University, Beijing 100084, China
(Dated: August 4, 2016)
We develop a theory for the quantum circuit consisting of a superconducting loop interrupted
by four Josephson junctions and pierced by a magnetic flux (either static or time-dependent). In
6 addition to the similarity with the typical three-junction flux qubit in the double-well regime, we
1 demonstrate the difference of the four-junction circuit from its three-junction analogue, including
0 its advantages over the latter. Moreover, the four-junction circuit in the single-well regime is also
2 investigated. Our theory provides a tool to explore the physical properties of this four-junction
superconductingcircuit.
g
u
A
SuperconductingquantumcircuitsbasedonJosephson son junctions in the four-junction circuit. We find that

3 junctions exhibit macroscopic quantum coherence and the four-junction circuit with only one smaller junction
can be used as qubits for quantum information process- has a broader parameter range to achieve a flux qubit

] ing (see, e.g., Refs. [1–9]). Behaving as artificial atoms, inthedouble-wellregimethanthethree-junctioncircuit.
h
these circuits can also be utilized to demonstrate novel Moreover,forthefour-junctioncircuitwithtwoidentical
p
- atomic-physics and quantum-optics phenomena, includ- smaller junctions, the circuitcan be used as a qubit bet-
t ing those that are difficult to observe or even do not oc- ter than the three-junction circuit, because it becomes
n
a cur in natural atomic systems [10]. As a rough distinc- more robust against the state leakage from the qubit
u tion,therearethreetypesofsuperconductingqubits,i.e., subspace to the third level. This can be a useful advan-
q charge[1,2],flux[4,11]andphasequbits[5,6,12]. Inthe tage of the four-junction circuit over the three-junction
[
charge qubit, where the charge degree of freedom domi- circuit when used as a qubit. Also, we study the four-

2 nates, two discrete Cooper-pair states are coupled via a junction circuit in the single-well regime, which was not
v Josephson coupling energy [1, 2]. In contrast, the phase exploited before. Our theory can provide a useful tool
2 degree of freedom dominates in both flux [11] and phase to explore the physical properties of this four-junction
7
qubits [5, 6]. superconducting circuit.
0
The typical flux qubit is composed of a superconduct-
0
ing loop interrupted by three Josephson junctions [11].
0
. Similar to other types of superconducting qubits, it ex-
2 Results
hibits good quantum coherence and can be tuned ex-
0
6 ternally. Recent experimental measurements [9] showed Four-junctionsuperconductingcircuit. (1)Theto-
1 thatthedecoherencetimeofthethree-junctionfluxqubit tal Hamiltonian. Let us considera superconducting loop
: canbelongerthan40µs. Duetotheconvenienceinsam- interrupted by four Josephson junctions and pierced by
v
ple fabrication (i.e., the double-layer structure fabrica- a magnetic flux [see Fig. 1(a)], where the first and sec-
i
X tion by the shadowevaporationtechnique [13]), a super- ond junctions have identical Josephson coupling energy
r conducting loopinterruptedby four Josephsonjunctions EJ and capacitance C (i.e., EJi =EJ and Ci =C, with
a was also used as the flux qubit. The experiments [14] i=1,2),whilethethirdandfourthjunctionsarereduced
showedthatthis four-junctionflux qubitbehavessimilar as E = αE , E = βE , C = αC, and C = βC,
J3 J J4 J 3 4
to the three-junction flux qubit. Also, two four-junction with 0 < α,β < 1. The phase drops ϕ (i = 1,2,3,4)
i
flux qubits were interacting experimentally via a cou- through these four Josephson junctions are constrained
pler [15], similar to the interqubit coupling mediated by by the fluxoid quantization
a high-excitation-energy quantum object [16]. The the-
ory of the three-junction flux circuit with a static flux
ϕ +ϕ +ϕ +ϕ +2πf (t)=0, (1)
bias was well developed [17], but a theory for the four- 1 2 3 4 tot
junctioncircuitlacksbecauseaddingoneJosephsonjunc-
tionmoretothesuperconductingloopmakestheproblem where ftot(t) = Φtot(t)/Φ0, with Φtot(t) being the to-
more complex. tal magnetic flux in the loop (which includes the exter-
Inthispaper,wedevelopatheoryforthefour-junction nally applied flux, either static or time-dependent, and
circuitwitheitherastaticortime-dependentfluxbias. In the inductance-induced flux owing to the persistent cur-
additiontothe similaritywiththe three-junctioncircuit, rentintheloop)andΦ0 =h/2ebeing the fluxquantum.
we demonstrate the difference from the three-junction The kinetic energy of the four-junction circuit is the
circuitduetothedifferentsizesofthetwosmallerJoseph- electrostaticenergy[18]storedinthejunctioncapacitors,
2
(a) φ E (b) where
1 J
2πβ
ξ =f f , b= ,
φ2EJ Φtot(t) φ4βEJ φ1EJ Φtot(t) φ2EJ tot− e −α+β+2αβ
α+β λ±
b± = | − | , (5)
2(α+β)2+4β2 4(β+α)λ±+2λ2±
φ αE φ αE −
3 J 3 J q
(1+α+3β) 1+(α β)2+8β2+2(β α)
λ± = ± − − ,
FIG. 1: (color online) Schematic diagram of the considered 2
p
superconducting circuits. (a) Superconducting loop inter-
with f =Φ /Φ being the reduced static magnetic flux
rupted by four Josephson junctions and pierced by a total e e 0
applied to the superconducting loop. The electrostatic
magnetic flux, Φ (t), which includes the externally applied
tot
energy can then be converted to a quadratic form
flux and the inductance-induced flux. Here two of the four
T
junctions have identical Josephson coupling energy EJ and 2
C Φ
csoanpaccoituapnlicnegCe.neArgmyoαngEJotahnedr tcwapoajcuintacnticoenαs,Co,naenhdasthJeosoetphher- T = 2 2π0 (ϕ˙2+Γ+ϕ˙2++Γ−ϕ˙2−+Γξξ˙2), (6)
(cid:18) (cid:19)
hasJosephsoncouplingenergyβEJ andcapacitanceβC,with
where
0<α,β <1. (b) Superconducting loop interrupted by three
Josephson junctions and pierced by a total magnetic flux 4β2 2β2(α+β)
Φpltiontg(te),newrghyerEeJtwaondjucnacptaiocnitsanhcaeveC,idwenhtiliecatlhJeotsheiprhdsoonnechoaus- Γ± =2b2±(cid:20)1+2β− α+β−λ± + (α+β−λ±)2(cid:21),
Josephson coupling energy αEJ and capacitance αC, with Γξ = 2α2+4βα2+α+β+4αβ b2 (7)
0<α<1. In both (a) and (b),each red component denotes
the thin insulator layer of a Josephson junction, and an ar- (cid:0)+4πβ(1+2α)b+4π2β. (cid:1)
rowalongtheloopdenotestheassigneddirectionofthephase
The total Josephson coupling energy of the four-
dropacrossthecorrespondingJosephson junction. Notethat
junction circuit is
each phase drop can be chosen along either the clockwise or
counter-clockwise direction, but once the direction is fixed, 4
thephase drop is positive along it. U = E (1 cosϕ )
Ji i
−
i=1
X
=E [2+α+β cosϕ cosϕ
J 1 2
− −
αcosϕ βcos(ϕ +ϕ +ϕ +2πf )]
which can be written as − 3− 1 2 3 tot
ϕ
=EJ 2+α+β cos +b+ϕ++b−ϕ−+αbξ
− √2
4 (cid:20) (cid:18) (cid:19)
1
= C V2, (2) ϕ
T 2Xi=1 i i −cos(cid:18)−√2 +b+ϕ++b−ϕ−+αbξ(cid:19)
2βb+ 2βb−
αcos ϕ+ ϕ−+bξ
where Vi =(Φ0/2π)ϕ˙i is the voltage acrossthe ith junc- − (cid:18)−α+β−λ+ − α+β−λ− (cid:19)
tion. Using the the fluxoid quantization condition in 2(α λ+)b+ 2(α λ−)b−
βcos − ϕ++ − ϕ−
Eq. (1), we can rewrite the kinetic energy as − (cid:18) α+β−λ+ α+β−λ−
+(2αb+b+2π)ξ+2πf . (8)
e
C Φ 2 (cid:19)(cid:21)
= 0 ϕ˙2+ϕ˙2+αϕ˙2 (3)
T 2 2π 1 2 3 Also,thereis the inductive energydue to the inductance
(cid:18) (cid:19) (cid:26) L of the superconducting loop [19]:
2
+β ϕ˙ +ϕ˙ +ϕ˙ +2πf˙ .
1 2 3 tot Φ2
h i (cid:27) UL = 0(ftot fext)2, (9)
2L −
where the reduced externally-applied magnetic flux f
We introduce a phase transformation ext
cangenerallybe written asa sumofthe static andtime-
dependent fluxes, i.e., f = f + f (t), with f (t)
ext e a a
≡
ϕ Φ (t)/Φ being the reduced time-dependent magnetic
ϕ1 = √2 +b+ϕ++b−ϕ−+αbξ, fieald app0lied to the four-junction loop. When including
ϕ this inductive energy, the total potential energy of the
ϕ2 =−√2 +b+ϕ++b−ϕ−+αbξ, (4) four-junction circuit is written as
2βb+ 2βb− Φ2
ϕ3 =−α+β λ+ϕ+− α+β λ−ϕ−+bξ, U =U(ϕ,ϕ+,ϕ−,ξ)+ 2L0(ξ−fa)2. (10)
− −
3
The Lagrangianof the four-junction circuit is i.e., a harmonic oscillator driven by a time-dependent
magnetic flux f (t). The angular frequency of this har-
a
= monic oscillator is
L T −U
= C2 Φ2π0 2(ϕ˙2+Γ+ϕ˙2++Γ−ϕ˙2−+Γξξ˙2) ωosc = 1 . (17)
(cid:18) (cid:19) ΓξCL
Φ2
U(ϕ,ϕ+,ϕ−,ξ) 0(ξ fa)2. (11) Withtheparametersachievpedinexperimentsfortheflux
− − 2L −
qubit [14, 20], α 0.7, C 8 fF, and L 10 pH. More-
∼ ∼ ∼
where we assign ϕ, ϕ±, and ξ as the canonical coor- over, β α, so ωosc/2π 1 103 GHz. For the four-
∼ ∼ ×
dinates. The corresponding canonical momenta P = junctionfluxqubit,theenergygap∆betweenthelowest
∂ /∂ϕ˙, P± =∂ /∂ϕ˙±, and Pξ =∂ /∂ξ˙ are two levels is typically ∆ 1-10 GHz [14, 15], which is
L L L much smaller than ω /2∼π 1 103 GHz. Usually,
osc
Φ0 2 the time-dependent magnetic∼flux×fa(t) applied to the
P =C ϕ˙,
four-junction loop is a microwave wave with ω /2π 1-
2π a
(cid:18) (cid:19) 10 GHz, which is also much smaller than ω /2π.∼Be-
P± =C Φ0 2Γ±ϕ˙±, (12) cause ∆ ≪ ωosc/2π and the flux fa(t) is aolssco very off
2π resonance from the harmonic oscillator (i.e., ω ω ),
(cid:18) (cid:19) a ≪ osc
2 the oscillator is nearly kept in the ground state at a low
Φ
Pξ =C 0 Γξξ˙. temperature. Then, using the adiabatic approximation
2π
(cid:18) (cid:19) to eliminate the degree of freedom of the oscillator, the
Hamiltonian of the four-junction circuit can be reduced
Therefore,theHamiltonianofthefour-junctioncircuitis
to
given by
P2 P2
H = P ϕ˙ H =4EC P2+ + + − +U(ϕ,ϕ+,ϕ−,ξ). (18)
i i−L (cid:18) Γ+ Γ−(cid:19)
i
X
P2 P2 P2 Also,bothLandthepersistentcurrentI ofthesupercon-
= 4E P2+ + + − + ξ ducting loop are small, so that IL/Φ 10−3 [17]. This
C 0
Γ+ Γ− Γξ! inductance-induced flux is much smalle∼r than the exter-
+U(ϕ,ϕ+,ϕ−,ξ)+ Φ20(ξ fa)2, (13) nthaellytoatpapllflieudxmfagnceatnicafllusoxbfeextap=prfoex+imfaat(etl)y. Twhrietrteefnoraes,
2L − tot
f f +f (t).
tot e a
≃
where E = e2/(2C) is the single-particle charging en- Below we first study the static-flux case, i.e., only a
C
ergy of the Josephson junction. In comparison with the static magnetic flux is applied to the four-junction loop.
previous work in Ref. [17] for the three-junctions flux In this case, ftot fe, so ξ 0. The phase transforma-
≃ ≃
qubit,anewdegreeoffreedomξisincludedintheHamil- tion in Eq. (4) becomes
tonian, so that the Hamiltonian can also apply to the
ϕ
case when the superconducting loop contains a time- ϕ1 = +b+ϕ++b−ϕ−,
√2
dependent magnetic flux.
ϕ
(2) The reduced Hamiltonian. The total Hamiltonian ϕ2 =−√2 +b+ϕ++b−ϕ−, (19)
of the four-junction circuit can be rewritten as 2βb+ 2βb−
ϕ3 = ϕ+ ϕ−,
P2 P2 −α+β−λ+ − α+β−λ−
H =4EC(cid:18)P2+ Γ++ + Γ−−(cid:19)+U(ϕ,ϕ+,ϕ−,ξ)+Hosc, and the Hamiltonian of the four-junction circuit in
(14) Eq. (18) is further reduced to
where
P2 P2
Hosc = 4ΓECPξ2+ 2ΦL20(ξ−fa)2, (15) H0 =4EC(cid:18)P2+ Γ++ + Γ−−(cid:19)+U(ϕ,ϕ+,ϕ−), (20)
ξ
with U(ϕ,ϕ+,ϕ−) U(ϕ,ϕ+,ϕ−,ξ)ξ=0.
≡ |
Quantum mechanically, the canonical momenta can be Figure 2 shows the contour plots of the potential
wrii~t∂te/n∂ϕaξsiPn t=he−cia~n∂o/n∂icϕa,l-Pco±or=din−ait~e∂r/e∂pϕre±s,enatnadtioPnξ. = sUu(bϕs,pϕa+ce,ϕs−pa)nn≡edUb(yϕ1ϕ,ϕ12a,nϕd3)ϕi2nfothrefetw=o-d1i/m2e,nwsihonerael
−Note that the Hamiltonian Hosc in Eq. (15) can be ϕi (i=1,2,3) are related to ϕ and ϕ± by Eq. (19). For
rewritten as a three-junction flux qubit, α is usually in the range of
1/2 < α < 1. When 0 < α < 1/2, each double well in
H = 4ECP2+ Φ20ξ2 Φ20ξf (t), (16) the potential is reduced to a single well [17], so the flux
osc Γ ξ 2L − L a qubit in the double-well regime is converted to a flux
ξ
4
(a) 2 (b)2 (c) 2
5.8 5.6 5.3
4.8 4.6 4.3
1 1 1
3.8 3.6 3.3
(cid:652) 2.8 2.6 2.3
(cid:21) 0 0 0
(cid:307)/2 1.8 1.6 1.3
-1 -1 -1
-2 -2 -2
-2 -1 0 1 2 -2 -1 0 1 2 -2 -1 0 1 2
(cid:307)1/(cid:21)(cid:652) (cid:307)1/(cid:21)(cid:652) (cid:307)1/(cid:21)(cid:652)
FIG.2: (coloronline)ContourplotsofthepotentialU(ϕ1,ϕ2,ϕ3)atϕ3 =0andfe =1/2,where(a)α=1,β=0.8,(b)α=1,
β =0.6, and (c) α=1, β =0.3.
qubitinthesingle-wellregime. Forthefour-junctioncir- where K is a reciprocal lattice vector. Substituting
cuit, there are wider ranges of parameters to achieve a Eq.(24)intoEq.(22),wethenobtainanequationsimilar
flux qubit. For instance, in the case of three identical tothecentralequationinthetheoryofenergybands[21].
Josephson junctions (i.e., α = 1 and 0 < β < 1), when Numerically solving this equation, we can obtainthe en-
β >1/3,thepotentialU(ϕ ,ϕ ,ϕ )hastwoenergymin- ergy spectrum and eigenstates of the Hamiltonian H .
1 2 3 0
ima in the unit cell of three-dimensional periodic lattice For the three-junction flux qubit, an approximate
∗
at ϕ =ϕ =ϕ = ϕ mod 2π, where tight-binding solution was obtained in Ref. [17] by pro-
1 2 3
±
jectingtheSchr¨odingerequationontothequbitsubspace,
∗ 3β 1 where the needed tunneling matrix elements were esti-
ϕ =arcsin − . (21)
s 4β ! mated using the WKB method. For the four-junction
case,suchanapproximatetight-bindingsolutioncanalso
A flux qubit in the double-well potential can then be be derived, but it is difficult to calculate the tunneling
achieved in the parameter range of 1/3 < β < 1, which matrix elements via the WKB method, because a three-
is broader than the range of 1/2 < α < 1 for the three- dimensionalpotentialisinvolvedinthefour-junctioncir-
junctionfluxqubit. Figures2(a)and2(b)showasection cuit. Thus, we resortto the numerical approachto solve
of U(ϕ1,ϕ2,ϕ3) at ϕ3 =0. Corresponding to the above- the Schr¨odinger equation in Eq. (22). With this numeri-
mentionedtwominima,afigure-eight-shapeddoublewell cal approach,we can obtain the results for both the flux
exists in each unit cell of the periodic lattice in the two- qubit and the three-level system.
dimensional subspace. When β <1/3, each figure-eight- Figure 3 shows the energy levels of the four-junction
shaped double well in the ϕ3 = 0 section of the poten- circuit versus the reduced static flux fe, in compari-
tial is reduced to a single well [see Fig. 2(c)], with only son with the three-junction circuit. In the case of four-
one minimum in the unit cell at ϕ1 = ϕ2 = 0 mod 2π. junction circuit, when the lowest two or three levels are
Thiscorrespondstoafluxqubitinthesingle-wellregime considered,the energy spectrum with α=1 and β =0.6
achieved in the four-junction superconducting circuit. is similar to the energy spectrum with α = 0.7 in the
case of three-junction circuit [comparing Fig. 3(c) with
Energy spectrum. The energy spectrum and eigen-
Fig. 3(a)]. Because the lowest two levels are well sepa-
states of the four-junction circuit are determined by
ratedfromotherlevels,boththree-andfour-junctioncir-
H Ψ(ϕ)=EΨ(ϕ), (22) cuits can be utilized as quantum two-level systems (i.e.,
0
flux qubits). In this case, the flux qubit can be modeled
where ϕ (ϕ,ϕ+,ϕ−) = (ϕ1,ϕ2,ϕ3) is a three- as
≡
dimensional vector in the phase space. Equation (22)
1
isjustlikethequantummechanicalproblemofaparticle
H = (εσ +∆σ ), (25)
0 z x
moving in a three-dimensional periodic potential U(ϕ). 2
Thus, the solution of it has the Bloch-wave form
where the tunneling amplitude ∆ correspondsto the en-
Ψ(ϕ)=eik·ϕu(ϕ), (23) ergy difference between the two lowest-energy levels at
f =1/2,andε=2I Φ (f 1/2)is the biasenergydue
e p 0 e
where k is a wavevector and u(ϕ) is a periodic function −
totheexternalflux,withI beingthemaximalpersistent
p
in the phases of ϕ (i = 1,2,3). Also, Ψ(ϕ) should be
i currentcirculatinginthe loop. Herethemaximalpersis-
periodic in the phases of ϕ . To ensure this, the wave-
i tent current I can be approximately calculated as [17]
p
function Ψ(ϕ) is constrained by k=0. Then, Ψ(ϕ) can I Φ−1∂E /∂f at a value of f considerably away
be written as p ≈ | 0 0 e| e
fromf =1/2,whereE istheenergyleveloftheground
e 0
Ψ(ϕ)=u(ϕ)= aKeiK·ϕ, (24) state of the system. The Pauli operators σz and σx are
representedusingthetwo(i.e.,theclockwiseandcounter-
K
X
5
FIG.3: (coloronline) Energyspectraof thesuperconductingcircuitsversusthereducedstaticfluxfe. (a)α=0.7and(b)0.4
inthecaseofthree-junctioncircuit;(c)α=1andβ=0.6,(d)α=1andβ =0.3,(e)α=β=0.6,(f)α=β =0.3,(g)α=0.5
and β =0.6, and (h) α=0.2 and β =0.3 in the case of four-junction circuit. In this figure and the following one, we choose
EJ =50EC
.
clockwise)persistent-currentstates. Moreover,similarto system. However, our calculations on transition matrix
the three-junction circuit, the four-junction circuit can elementsindicate thatthe circuitcanstillbe better used
also be used as a quantum three-level system (qutrit) as a qubit, because only the transition matrix element
owing to the considerable separation of the third energy between the ground and first excited states is apprecia-
levelfromotherhigherlevelsaswell. Whenreducingthe bly large (see the next section).
smallest junction to, e.g., β = 0.3 in the four-junction
In addition, we further consider the case of two differ-
circuit [see Fig. 3(d)], only the lowest two levels are well
ent smallerJosephsonjunctions (i.e., α=β) in the four-
separated from other levels, similar to the case of three- 6
junctioncircuit. Inthe double-wellregime[seeFig.3(g),
junction circuit in Fig. 3(b) where α = 0.4. Now the
where α = 0.5 and β = 0.6], the energy levels look sim-
double-well potential has been convertedto a single well
ilar to those in Fig. 3(e) and the lowest two levels can
(see Fig. 2), so the circuit behaves as a flux qubit in
still be used as a qubit. Also, this qubit is less sensitive
the single-well regime. Compared to the flux qubits in
to the influence of the external magnetic field around
Figs. 3(a) and 3(c), the energy levels in Figs. 3(b) and
the degeneracypoint,becausethe energylevelsaremore
3(d) are less sensitive to the external flux f , so the ob-
e flat than those in Fig. 3(e). In the single-well regime
tained flux qubits in the single-well regime are more ro-
[see Fig. 3(h), where α = 0.2 and β = 0.3], the low-
bust against the flux noise. However,because the small-
est three levels are well separatedfrom the higher levels.
estJosephsonjunctionintheloopisfurtherreduced,the
Moreover, in addition to the transition matrix element
charge noise may become important [22]. To suppress
between the ground and first excited states, the transi-
this charge noise, one can shunt a large capacitance to
tionmatrixelementbetweenthe firstandsecondexcited
the smallest junction to improve the quantum coherence
states is also larger (see the section below). Therefore,
of the qubit [9, 22, 23].
in the single-well regime, the four-junction circuit in the
Furthermore, let us consider the four-junction cir- caseofα=β canbebetterusedasaquantumthree-level
6
cuit with two identical smaller Josephson junctions. In system. ThisisdifferentfromthecasesinFigs.3(b),3(d)
Fig. 3(e) where α = β = 0.6, the lowest two levels are and 3(f).
also well separated from other levels, but the third level
is not so separated from higher levels. Thus, from the Transition matrix elements. Now we consider the
energy-level point of view, this four-junction circuit can time-dependent case with f (t) f + f (t), i.e., in
tot e a
≃
be better used as a flux qubit than a three-level system. additiontoastaticfluxf ,atime-dependentfluxf (t)
e a
≡
In Fig. 3(f) where α = β = 0.3, the lowest three levels Φ (t)/Φ isalsoappliedtothefour-junctionloop. Inthis
a 0
are well separated from other levels. It seems that the case, ξ f (t) when ignoringthe very smallinductance-
a
≃
four-junction circuit can be better used as a three-level induced flux. For a small enough time-dependent flux,
6
only the first-order perturbation due to ξ needs to be Here we consider a microwave field with frequency
considered in Eq. (18). Then, the Hamiltonian of the ω applied to the superconducting loop. The time-
a
four-junction circuit in Eq. (18) can be expressed as dependent magnetic flux in the loop can be written as
Φ (t)=Φ(0)cosω t. Then, with the current I available,
′ a a a
H =H +H (t), (26)
0 the magnetic-dipole transition matrix elements are cal-
culated by
with H given in Eq. (20) and
0
t = iIΦ(0) j , (32)
′ ϕ ij h | a | i
H (t)=fa(t)EJ αbsin +b+ϕ++b−ϕ−
√2 where i and j are eigenstates of the Hamiltonian H
(cid:20) (cid:18) (cid:19) | i | i 0
ϕ in Eq. (20).
+αbsin −√2 +b+ϕ++b−ϕ− Figure 4 shows the transition matrix elements |t01|,
(cid:18) (cid:19) t , and t of the three- and four-junction circuits as
02 12
αbsin 2βb+ ϕ++ 2βb− ϕ− |a fu|nction|of|the reduced static flux fe, where the sub-
− (cid:18)α+β−λ+ α+β−λ− (cid:19) scripts0,1 and2 correspondto the groundstate|0i, the
2(α λ )b first excited state 1 , and the second excited state 2
+(2αb+b+2π)βsin − + +ϕ | i | i
α+β λ + ofthe system, respectively. Similar to the three-junction
(cid:18) − + circuit in Fig. 4(a) where α = 0.7, the four-junction cir-
2(α λ−)b−
+ − ϕ−+2πfe . (27) cuit with α = 1 and β = 0.6 (i.e., there is only one
α+β−λ− (cid:19)(cid:21) smaller Josephson junction in the circuit) behaves as a
Thetime-dependentperturbationH′(t)canberewritten ladder-type (namely, Ξ-type [24]) three-level system at
f =1/2,and a cyclic-type (∆-type [25]) three-levelsys-
as e
tematf =1/2[seeFig.4(c)]. FortheΞ-typethree-level
e
6
H′(t)= IΦa(t), (28) system achieved when fe = 1/2, the transition between
− the ground state 0 and the second excited state 2 is
| i | i
where notallowed,whichis analogousto anaturalatom. How-
ever, for the ∆-type three-level system at f = 1/2, all
I =−EΦJ0(cid:20)αbsin(cid:18)√ϕ2 +b+ϕ++b−ϕ−(cid:19) dtriaffnesrietniotnfsroammoannga|t0uir,al|1aitoamndic|2syisaterema[l2lo5w].eed6W. hTenhisthies
ϕ smallest Josephson junction is further reduced, t is
+αbsin −√2 +b+ϕ++b−ϕ− greatly suppressed. Now both three- and four-ju|n0c2t|ion
(cid:18) (cid:19)
circuits behave more like a Ξ-type three-level system in
2βb+ 2βb−
−αbsin(cid:18)α+β−λ+ϕ++ α+β−λ−ϕ−(cid:19) theAws hfoolre trhegeiofnouorf-jfuenschtioownnciinrcFuiigts.w4i(tbh) tawnod 4id(de)n.tical
+(2αb+b+2π)βsin 2(α−λ+)b+ϕ+ smaller Josephson junctions (α=β), while |t01| remains
α+β λ appreciably large, the transition between 0 and 2 as
(cid:18) − + | i | i
+ 2α(α+−βλ−λ)−b−ϕ−+2πfe (29) dwueclledas(tih.ee.,tr|at0n2s|it≈ion0baetnwdee|tn12|1|i≈an0d)|2ini atrheegwrehaotlley rree--
− (cid:19)(cid:21) gion of f shown in Figs. 4(e) and 4(f). Now, in either
e
is the current in the superconducting loop. Because double- or single-well regime, the four-junction circuit
can be well used as a qubit, because the state leakage
(2αb+b+2π)β = αb from the qubit subspace to the third level is suppressed.
−
2παβ Thisisanapparentadvantageofthefour-junctioncircuit
= , (30)
−α+β+2αβ over the three-junction circuit when used as a qubit.
WhenthetwosmallerJosephsonjunctionsinthefour-
we can express the current I as junction circuit become different (i.e., α = β), in ad-
6
dition to t , both t and t become nonzero ex-
01 02 12
αβ 2πE | | | | | |
I = J [sinϕ +sinϕ +sinϕ cept for the degeneracy point [see Figs. 4(g) and 4(h)].
1 2 3
α+β+2αβ (cid:18) Φ0 (cid:19) This circuit behaves very different from the circuit with
sin(ϕ +ϕ +ϕ +2πf )] twoidenticalsmallerjunctions[comparingFig.4(g)with
1 2 3 e
−
1 Fig. 4(e), and comparing Fig. 4(h) with Fig. 4(f)], but
= (αβI +αβI +βI +αI ), (31)
1 2 3 4 it is similar to the three-junction circuit and the four-
α+β+2αβ
junction circuit with only one smaller junction [compar-
where I = I sinϕ , with i = 1,2 and I = 2πE /Φ , ing Fig. 4(g) with Figs. 4(a) and 4(c), and comparing
i c 1 c J 0
I =αI sinϕ , andI =βI sinϕ areJosephsonsuper- Fig. 4(h) with Figs. 4(b) and 4(d)]. However, when the
3 c 3 4 c 4
currents through the four junctions. The phase drops ϕ distributionoftheenergylevelsisalsotakenintoaccount
i
(i=1,2,3) are related to ϕ and ϕ± by Eq. (19), and ϕ4 (see Fig. 3), the four-junction circuit with α= β can be
6
is constraintby the fluxoidquantizationconditioninthe better used as a quantum three-level system (qutrit) in
static-flux case, i.e., ϕ +ϕ +ϕ +ϕ +2πf =0. the single-well regime. This is very different from the
1 2 3 4 e
7
0.6 0.6 0.6 0.6
(a) |t | (c) (e) (g)
01
|t |
02
0.4 |t | 0.4 0.4 0.4
12
0.2 0.2 0.2 0.2
s
nt
e
m
ele 0.0 0.0 0.0 0.0
x
atri 0.48 0.49 0.50 0.51 0.52 0.48 0.49 0.50 0.51 0.52 0.48 0.49 0.50 0.51 0.52 0.48 0.49 0.50 0.51 0.5.2
m
sition 0.3 (b) 0.3 (d) 0.3 (f) 0.3 (h)
n
a
Tr 0.2 0.2 0.2 0.2
0.1 0.1 0.1 0.1
0.0 0.0 0.0 0.0
0.48 0.49 0.50 0.51 0.52 0.48 0.49 0.50 0.51 0.52 0.48 0.49 0.50 0.51 0.52 0.48 0.49 0.50 0.51 0.52
fe fe fe fe
FIG.4: (coloronline)Transition matrixelements|t01|,|t02|and|t12|ofthesuperconductingcircuits(inunitsofIcΦ(a0))versus
the reduced static flux fe. (a) α=0.7 and (b) 0.4 in the case of three-junction circuit; (c) α=1 and β =0.6, (d) α=1 and
β =0.3, (e) α=β =0.6, (f) α=β =0.3, (g) α=0.5 and β =0.6, and (h) α=0.2 and β =0.3 in the case of four-junction
circuit.
three-junction circuit and the four-junction circuit with regimes. In addition to the similarity with the three-
only one smaller junction, which can be better used as a junction circuit, we show the difference of the four-
qubit in the single-well regime. Therefore, as compared junction circuit from its three-junction analogue. Also,
tothethree-junctioncircuit,thefour-junctioncircuitcan we demonstrate its advantages over the three-junction
provide more choices to achieve different quantum sys- circuit. Owing to the one additional Josephson junction
tems. inthecircuit,thephysicalpropertiesofthefour-junction
circuitbecomericherthanthoseofthethree-junctioncir-
cuit. For instance, in the case of four-junction circuit
withonlyonesmallerJosephsonjunction,thecircuithas
Summary
a broader parameter range to achieve a flux qubit in the
We have developed a theory for the four-junction super- double-well regime than the three-junction circuit does.
conducting loop pierced by an externally applied mag- Moreover, in the case of four-junction circuit with two
netic flux. When the loop inductance is considered, the identical smaller junctions, the circuit can be used as
derived Hamiltonian of this four-junction circuit can be a qubit better than the three-junction circuit in both
written as the sum of two parts, one of which is the double- and single-well regimes. This is because among
Hamiltonian of a harmonic oscillator with a very large the lowest three eigenstates of the four-junction circuit,
frequency. This makes it feasible to employ the adia- only the transition matrix element between the ground
batic approximation to eliminate the degree of freedom and first excited states is appreciably large, while other
oftheharmonicoscillatorinthetotalHamiltonian. Also, two elements become zero. These properties of the four-
this theory can be used to study the case when the ap- junction circuit can suppress the state leakage from the
pliedmagnetic-fluxbiasbecomestime-dependent. Inthe qubit subspace to the second excited state, and the cir-
case of static flux bias, the total Hamiltonian of the cuitwiththeseparametersisthusexpectedtohavebetter
four-junction circuit is reduced to the Hamiltonian of quantum coherence when used as a qubit.
the superconducting qubit. When the flux bias is time-
dependent,thetotalHamiltonianofthefour-junctioncir-
cuit can be reduced to the Hamiltonian of the supercon-
Methods
ducting qubit plus a perturbation related to the applied
time-dependent flux. Then, we can calculate the energy Three-junction circuit with a time-dependent
spectrumandthe transitionmatrixelements ofthe four- magnetic flux. To compare with our four-junction re-
junction superconducting circuit. sults, we also consider a three-junction superconducting
In conclusion, we have studied the four-junction su- loop pierced by a time-dependent total magnetic flux
perconducting circuit in both double- and single-well Φ (t) [see Fig. 1(b)], because no explicit derivation ex-
tot
8
ists in the literature for this time-dependent case. The Because L is small in a three-junction flux qubit [17],
directions of the phase drops ϕ (i = 1,2,3) through we canignorethe flux generatedby the loopinductance.
i
the three Josephson junctions are chosen as in Ref. [17], Thus, when only a static flux is applied to the loop,
which are constrained by the following fluxoid quantiza- f (t) f , i.e., ξ 0. The phase transformation in
tot e
≃ ≃
tion condition: Eq. (34) becomes
ϕ1−ϕ2+ϕ3+2πftot(t)=0, (33) ϕ = ϕ1+ϕ2, ϕ = ϕ1−ϕ2, (40)
p m
2 2
where f (t) = Φ (t)/Φ . Here we assume that two
tot tot 0
larger junctions have identical capacitance C and cou-
andthe Hamiltonianof the circuitin Eq.(39)is reduced
pling energy E , while the smaller junction has capaci-
J to
tance αC and coupling energy αE , with 0<α<1.
J
Similar to the four-junction circuit, we introduce a
2E
phase transformation H0 =2ECPp2+ (1+C2α)Pm2 +EJ[2+α (41)
ϕp = ϕ1+ϕ2, −2cosϕpcosϕm−αcos(2ϕm+2πfe)],
2
ϕ ϕ 2πα
1 2 whichisthe Hamiltonianofthe three-junctionfluxqubit
ϕ = − + ξ, (34)
m
2 1+2α derived in Ref. [17].
For the time-dependent case with f (t) f +f (t),
where ξ ftot(t) fe, with fe = Φe/Φ0 being the re- tot ≃ e a
≡ − ξ f (t), where f (t) Φ (t)/Φ is the reduced time-
duced static magnetic flux applied to the superconduct- ≃ a a ≡ a 0
dependent magnetic flux applied to the three-junction
ing loop. The Hamiltonian of the three-junction circuit
loop. When the time-dependent magnetic flux is small
can be derived as
enough, only the first-order perturbation due to ξ needs
H =2E P2+ 2EC P2 +U(ϕ ,ϕ ,ξ)+H , (35) to be considered, and the Hamiltonian of the circuit in
C p (1+2α) m p m osc Eq. (39) can be expressed as
where EC =e2/(2C), H =H +H′(t), (42)
0
2πα
U =EJ 2+α−2cosϕpcos ϕm− 1+2αξ with H0 given in Eq. (41) and H′(t)=−IΦa(t), where
(cid:26) (cid:18) (cid:19)
2π
−αcos 2ϕm+ 1+2αξ+2πfe , (36) I = 2πα EJ [2cosϕpsinϕm sin(2ϕm+2πfe)]
(cid:18) (cid:19)(cid:27) 1+2αΦ0 −
(43)
and
is the current in the three-junction loop [26]. Using
E (1+2α) Φ2 Eq. (40) and the fluxoid quantization condition in the
H = C P2+ 0(ξ f )2. (37)
osc π2α ξ 2L − a static-flux case (i.e., ϕ1 ϕ2+ϕ3+2πfe = 0), the cur-
−
rent I can also be rewritten as
Quantum mechanically, the canonical momenta can be
written as P = i~∂/∂ϕ , P = i~∂/∂ϕ , and P =
p p m m ξ α 2πE
−iT~∂h/e∂aϕnξguinlarthfre−ecqauneonnciycaolf-cthooerhdainr−matoenriecporsecsiellnattaotriogniv.en I = 1+2α(cid:18) Φ0J(cid:19)[sinϕ1−sinϕ2
sin(ϕ ϕ +2πf )]
in Eq. (37) is 1 2 e
− −
1
= (αI αI +I ), (44)
1+2α 1+2α 1− 2 3
ω = . (38)
osc
αCL
r
where I is the Josephson supercurrent through each
i
Using the parameters achieved in experiments [14, 20], junction. Moreover, as in Eq. (32), the magnetic-
we have α 0.7, C 8 fF, and L 10 pH, so one dipole transitionmatrix elements are calculatedby t =
has ωosc/2π∼∼ 103 GH∼z, which is muc∼h larger than the iIΦ(a0) j ,where i and j areeigenstatesoftheHaimjil-
energygap∆ 1-10GHzofthethree-junctionfluxqubit h | | i | i | i
(see, e.g., Ref.∼[4]). If the time-dependent magnetic flux tonian H0 in Eq. (41).
is the usually applied microwave field, the oscillator can
indeed be regardedas being in the groundstate at a low
temperature,asanalyzedforthe four-junctionfluxqubit Acknowledgement
in the main text. Then, the Hamiltonian of the three- This work is supported by the NSAF Grant No.
junction circuit can be reduced to U1330201, the National Natural Science Foundation of
China Grant No. 91121015, and the National Basic Re-
2E
H =2E P2+ C P2 +U(ϕ ,ϕ ,ξ). (39) search Programof China Grant No. 2014CB921401.
C p (1+2α) m p m
9
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